1. **Problem statement:** A ship travels on a N 500 E course (meaning 50 degrees east of north). It travels until it is due north of a port that is 10 km due east of the starting port. We need to find how far the ship traveled. 2. **Understanding the problem:** The ship starts at the origin (0,0). The port it must be north of is at (10,0) since it is 10 km east of the start. 3. **Course direction:** N 500 E means the ship's path makes a 50° angle east of north. In standard coordinate terms, north is along the positive y-axis, so the ship's direction vector forms a 50° angle with the y-axis towards the east. 4. **Parametric equations of the ship's path:** Let the distance traveled be $d$. The ship's coordinates after traveling distance $d$ are: $$x = d \sin 50^\circ$$ $$y = d \cos 50^\circ$$ 5. **Condition for being due north of the port at (10,0):** The ship must be directly north of (10,0), so its x-coordinate must be 10: $$d \sin 50^\circ = 10$$ 6. **Solve for $d$:** $$d = \frac{10}{\sin 50^\circ}$$ 7. **Calculate $d$:** Using $\sin 50^\circ \approx 0.7660$, $$d \approx \frac{10}{0.7660} \approx 13.06$$ **Final answer:** The ship traveled approximately 13.06 kilometers.